A reliable graphical criterion for TDS stability analysis

The paper considers the stability issue of linear systems with commensurate delays. This issue can be well characterised by the distribution of roots of system's characteristic equation. At first, distribution boundary of the roots (with positive real parts) is explicit given in a practical way. Subsequently, a reliable graphical stability criterion for calculating the number of unstable roots is deduced, associating with auxiliary polynomial which plays an important role in the analysis of high order and complex systems. Moreover, a procedure for drawing the winding curve of characteristic function in finite path is proposed. At last, typical examples are given to illustrate that the result carried out is reliable and efficient.

[1]  Allan M. Krall.  Stability Criteria for Feedback Systems with a Time Lag , 1964 .

[2]  Nejat Olgac,et al.  Stability of formation control using a consensus protocol under directed communications with two time delays and delay scheduling , 2016, Int. J. Syst. Sci..

[3]  Donald F. Towsley,et al.  Analysis and design of controllers for AQM routers supporting TCP flows , 2002, IEEE Trans. Autom. Control..

[4]  L. K. Barker.  Mikhailov Stability Criterion for Time-delayed Systems , 1979 .

[5]  C. Hsu,et al.  On the $\tau $-Decomposition Method of Stability Analysis for Retarded Dynamical Systems , 1969 .

[6]  Jie Chen,et al.  On stability crossing curves for general systems with two delays , 2004 .

[7]  Suchin Arunsawatwong.  Stability of retarded delay differential systems , 1996 .

[8]  Jianbin Qiu,et al.  Adaptive Fuzzy Control for Nontriangular Structural Stochastic Switched Nonlinear Systems With Full State Constraints , 2019, IEEE Transactions on Fuzzy Systems.

[9]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[10]  Jianbin Qiu,et al.  Observer-Based Fuzzy Adaptive Event-Triggered Control for Pure-Feedback Nonlinear Systems With Prescribed Performance , 2019, IEEE Transactions on Fuzzy Systems.

[11]  Silviu-Iulian Niculescu,et al.  On Computing Puiseux Series for Multiple Imaginary Characteristic Roots of LTI Systems With Commensurate Delays , 2013, IEEE Transactions on Automatic Control.

[12]  Meng Wang,et al.  Finite Frequency Memory Output Feedback Controller Design for T–S Fuzzy Dynamical Systems , 2018, IEEE Transactions on Fuzzy Systems.

[13]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[14]  Young Soo Suh,et al.  Stability of time delay systems using numerical computation of argument principles , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[15]  K. Cooke,et al.  On zeroes of some transcendental equations , 1986 .

[16]  Kirk S. Walton,et al.  Direct method for TDS stability analysis , 1987 .

[17]  Jianbin Qiu,et al.  A Switched System Approach to Exponential Stabilization of Sampled-Data T–S Fuzzy Systems With Packet Dropouts , 2016, IEEE Transactions on Cybernetics.

[18]  Emilia Fridman,et al.  Tutorial on Lyapunov-based methods for time-delay systems , 2014, Eur. J. Control.

[19]  Roman Prokop,et al.  A Stability Test for Control Systems with Delays Based on the Nyquist Criterion , 2011 .

[20]  T. Kerlin,et al.  An Efficient, Computer-Oriented Method for Stability Analysis of Large Multivariable Linear Systems , 1970 .

[21]  Rifat Sipahi,et al.  An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems , 2002, IEEE Trans. Autom. Control..

[22]  B. Hassard,et al.  Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems , 1997 .

[23]  Peilin Fu,et al.  An Eigenvalue Perturbation Approach to Stability Analysis, Part I: Eigenvalue Series of Matrix Operators , 2010, SIAM J. Control. Optim..

[24]  Wim Michiels,et al.  Invariance properties in the root sensitivity of time-delay systems with double imaginary roots , 2010, Autom..

[25]  Tomás Vyhlídal,et al.  Modification of Mikhaylov Criterion for Neutral Time-Delay Systems , 2009, IEEE Transactions on Automatic Control.

[26]  Sabine Mondié,et al.  Robust stability of quasi-polynomials and the finite inclusions theorem , 2005, IEEE Transactions on Automatic Control.

[27]  Keqin Gu,et al.  Stability and Stabilization of Systems with Time Delay , 2011, IEEE Control Systems.

[28]  Jong-Tae Lim,et al.  On stability of linear time-delay systems with multiple delays , 2008, Int. J. Syst. Sci..

[29]  Simone Baldi,et al.  Adaptive stabilization of impulsive switched linear time-delay systems: A piecewise dynamic gain approach , 2019, Autom..

[30]  A reliable stability test for exponential polynomials , 1992 .

[31]  Fei Shu-min,et al.  Stability of Linear Systems with Time Delay: A New Delay Fractioning Approach , 2006, 2007 Chinese Control Conference.